An exact method for solving the problem of minimizing the maximum of a finite number of functions consists of solving a sequence of sub problems when quadratic approximations to the functions are employed in the determination of a search direction. For problems of large size, solving the sub problems exactly can be very expensive. In this paper we study truncated methods for solving the minimax problem. In such a truncated method, the sub problems and quadratic sub problems are solved only up to a certain degree of accuracy. The necessary accuracies that are needed to preserve the nice features of the exact method are established. The numerical results show that this method is efficient.